Algebraic General Topology is a certain theory that generalizes general topology (both point-set topology and pointfree topology) in an algebraic way.
There are three books available:
- Algebraic General Topology: Book 1: Basics
- Algebraic General Topology: Book 3: Algebra
- Limit of a Discontinuous Function
The "Basics" book generalizes together topological spaces, pretopological spaces, closure spaces, (quasi-)proximity spaces, (directed) graphs (=binary relations) into one concept of funcoids, and both uniform spaces and (directed) graphs (=binary relations) are generalized as reloids, that is filters on binary Cartesian products.
That funcoids and reloids are generalizations of both spaces and functions makes them convenient for study of functions on spaces. For example, continuity is defined by simple algebraic formulas rather than by epsilon-delta notation.
The books further considers pointfree and multidimensional generalizations, various products, etc.
The book 3 "Algebra" generalizes it even further. It shows that every space in topology (all above mentioned spaces, frames, locales, and even metric spaces) is an element of an ordered semigroup action (that is an element of a semigroup action conforming to a partial order). Thus the entire general topology is just ordered semigroup.
And finally "Limit of a Discontinuous function" uses the above theory to define a generalization of limit that is applicable to every (even discontinuous) function. This makes obvious to define derivative and integrals of arbitrary functions. Derivative and integrals as defined in the book are still linear operators! The book considers also some basic properties of differential equations with generalized derivatives.
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